Verify the Identity csc(x)=cot(x)sec(x)

Problem

csc(x)=cot(x)*sec(x)

Solution

1. Identify the right side of the identity to manipulate, as it contains more complex trigonometric functions.

2. Apply the definitions of the cotangent and secant functions in terms of sine and cosine.

cot(x)=cos(x)/sin(x)

sec(x)=1/cos(x)

3. Substitute these definitions into the expression for the right side.

cot(x)*sec(x)=cos(x)/sin(x)⋅1/cos(x)

4. Simplify the expression by canceling the common factor of cos(x) from the numerator and the denominator.

cos(x)/(sin(x)*cos(x))=1/sin(x)

5. Recognize the resulting expression as the definition of the cosecant function.

1/sin(x)=csc(x)

Final Answer

csc(x)=cot(x)*sec(x)

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